Polygon Angles Worksheets with Answers | KS3 - Free Printable

Let’s solve each part step by step.

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Question 1:
We need an expression for the sum of interior angles in any n-sided polygon.

✔ Rule: For any polygon with n sides, the sum of interior angles is:

> (n - 2) × 180°

Why? Because you can split any polygon into (n - 2) triangles, and each triangle has 180°.

So answer for Q1:
→ (n - 2) × 180

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Now let’s do questions 2 to 6. We’ll fill in the table columns:

- Name of Shape → based on number of sides
- Number of sides → count them
- Sum of all angles → use formula above
- Value of missing angle x° → subtract known angles from total

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Question 2: Triangle

Shape: Triangle → 3 sides

Sum of angles = (3 - 2) × 180 = 180°

Given angles: 33°, 71°, and x°

So:
x = 180 - 33 - 71 = 76°

Check: 33 + 71 + 76 = 180 ✔

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Question 3: Quadrilateral

Shape: Quadrilateral → 4 sides

Sum of angles = (4 - 2) × 180 = 360°

Given angles: 74°, 112°, 96°, and x°

Add known: 74 + 112 + 96 = 282

x = 360 - 282 = 78°

Check: 74 + 112 + 96 + 78 = 360 ✔

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Question 4: Pentagon

Shape: Pentagon → 5 sides

Sum of angles = (5 - 2) × 180 = 540°

Given angles: 101°, 168°, 126°, 89°, and x°

Add known: 101 + 168 = 269; 269 + 126 = 395; 395 + 89 = 484

x = 540 - 484 = 56°

Check: 101+168+126+89+56 = 540 ✔

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Question 5: Hexagon

Shape: Hexagon → 6 sides

Sum of angles = (6 - 2) × 180 = 720°

Given angles: 155°, 163°, 92°, 112°, and one right angle (90°), plus x°

Wait — look at diagram: there are 6 angles shown.

List them:

- 155°
- x°
- 163°
- 92°
- 112°
- and a right angle symbol → that’s 90°

So known angles: 155 + 163 + 92 + 112 + 90 = ?

Calculate:

155 + 163 = 318
318 + 92 = 410
410 + 112 = 522
522 + 90 = 612

Total sum needed: 720°

x = 720 - 612 = 108°

Check: 155+163+92+112+90+108 = 720 ✔

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Question 6: Octagon?

Count the sides/angles in the shape.

Looking at the diagram: it's a complex irregular polygon with 8 vertices → so octagon

Number of sides = 8

Sum of angles = (8 - 2) × 180 = 6 × 180 = 1080°

Now list all given angles:

From diagram:

- 95°
- 209° ← this is a reflex angle? But in polygons, we usually take interior angles as less than 180° unless specified. Wait — but here it’s drawn as going inward? Actually, looking again — this might be a concave polygon, and some angles are reflex (greater than 180°). In such cases, we still use the same formula — the sum is still (n-2)*180, even if some angles are reflex.

But wait — let’s check the diagram carefully.

Actually, looking at question 6: it shows a star-like or indented shape with labeled angles including 209°, 250°, etc. These are likely the interior angles, even if they’re reflex.

In standard geometry problems like this (especially GCSE level), when angles are marked inside the shape—even if greater than 180°—they are considered interior angles, and the formula still applies.

So proceed.

Angles given:

- 95°
- 209°
- 134°
- 77°
- 84°
- 126°
- 250°
- 106°
- 87°
- and x°

Wait — how many angles total?

Let me count the labeled angles in the diagram for Q6:

There are 10 angles labeled? That would mean it’s a decagon? Let me recount.

Actually, looking again — perhaps I miscounted.

Better approach: Count the number of sides by counting the line segments forming the boundary.

But since it’s messy, let’s count the number of angle labels:

In Q6 diagram:

Top left: 95°
Then moving clockwise: 209°, 134°, 77°, 84°, 126°, 250°, 87°, 106°, and then x° near bottom left.

That’s 10 angles → so it’s a decagon (10 sides)

Confirm: 10 sides → sum = (10 - 2) × 180 = 8 × 180 = 1440°

Now add up all given angles except x:

List:

95
209
134
77
84
126
250
87
106

Add step by step:

Start: 95 + 209 = 304
304 + 134 = 438
438 + 77 = 515
515 + 84 = 599
599 + 126 = 725
725 + 250 = 975
975 + 87 = 1062
1062 + 106 = 1168

So sum of known angles = 1168°

Total should be 1440°

x = 1440 - 1168 = 272°

Wait — 272°? That seems very large, but possible for a reflex interior angle in a concave polygon.

Double-check addition:

95
209 → 304
134 → 438
77 → 515
84 → 599
126 → 725
250 → 975
87 → 1062
106 → 1168 ✔

1440 - 1168 = 272 ✔

So x = 272°

Even though it’s large, mathematically correct for this context.

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Now compile answers for the table:

| Q | Name of Shape | Number of sides | Sum of all angles | Value of missing angle x° |
|---|-------------------|------------------|--------------------|----------------------------|
| 2 | Triangle | 3 | 180° | 76° |
| 3 | Quadrilateral | 4 | 360° | 78° |
| 4 | Pentagon | 5 | 540° | 56° |
| 5 | Hexagon | 6 | 720° | 108° |
| 6 | Decagon | 10 | 1440° | 272° |

And Q1: Expression is (n - 2) × 180

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Final Answer:

1) (n - 2) × 180
2) Triangle, 3, 180°, 76°
3) Quadrilateral, 4, 360°, 78°
4) Pentagon, 5, 540°, 56°
5) Hexagon, 6, 720°, 108°
6) Decagon, 10, 1440°, 272°